On the calculation of unique force constants
Abstract
The vibrational F matrix can be written in the parameterized form F = L_{0}^{+1}C^{+1} Λ C^{1}L_{0}^{1}, where L_{0} is determined from the G matrix alone; Λ is from the vibration frequencies; and C is an orthogonal matrix of {N(N1)}/{2} parameters φ_{ij}. The restoring forces for internal coordinates or symmetry coordinates are calculated as a function of the φ_{ij} parameters. Conditions on the restoring forces and energy are proposed that can be summarized in the equation F _{steep} = Σ_{i} T _{i}D _{ii}^{ {1}/{2}}. The critical points of F_{stp} or certain "intersection" points (e.g., T _{1}F _{22}^{ {1}/{2}} = T _{2}F _{11}^{ {1}/{2}} for N = 2) determine the several mathematically acceptable sets of φ_{ij} parameters. The molecular "true" F matrix has nearmaximum parameterized restoring forces for coordinates associated with the highest vibration frequencies. In the F_{stp} equation, the T_{i} are geometrical factors derived using the virial theorem (e.g., T_{1} = r and T_{2} = 2 cosθ for C_{2 v} triatomic molecules) and the D_{ii} are cofactors of the diagonal force constants. The method is illustrated with two, three and fourdimensional examples including F _{2}O, NF _{3}, ONF, H _{2}S _{2} and other molecules. The force constants determined using the new method with the vibration frequencies of a single molecule compare very well with those determined by conventional methods using isotopic substitution, Coriolis coupling constants, and centrifugal distortion constants. For many molecules the F matrix is determined uniquely by knowledge of the vibrational assignment, since only one F matrix solution occurs with the proper assignment; in other cases, a certain minimal amount of additional information beyond the vibration frequencies and geometry may be necessary in order to select the "true" F matrix. For example, the D_{JK} centrifugal distortion constants can isolate the true F matrices for the nitrosyl halides.
 Publication:

Journal of Molecular Spectroscopy
 Pub Date:
 March 1971
 DOI:
 10.1016/00222852(71)901792
 Bibcode:
 1971JMoSp..37..494R